Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 67 / 3 / ii / Solution

For $\Lambda =0$, use the instantaneous cosmological density parameterInsert the Friedmann equation from the preceding part and multiply numerator and denominator by $a^4$:This holds on an expanding or contracting branch with $H\ne 0$; in a recollapsing model the density parameter diverges at turnaround.

During radiation domination, while curvature is small, $|\Omega -1|\simeq |{\Omega }_0-1|a^2/{\Omega }_{R0}$. During matter domination the corresponding behavior is $|\Omega -1|\simeq |{\Omega }_0-1|a/{\Omega }_{M0}$. Thus the exactly flat solution $\Omega =1$ is unstable against relative curvature during decelerating expansion: a small departure grows rather than being dynamically driven to zero. The flatness problem is that a universe with modest present curvature must have had an extremely tiny initial curvature-to-density ratio at an early radiation epoch. The fact that every such solution tends to $\Omega =1$ when extrapolated to $a\to 0$ does not supply a mechanism selecting that small departure at a specified finite initial epoch. An early period of cosmic inflation can instead shrink the comoving Hubble radius and suppress $|\Omega -1|=|k|/(aH{)}^2$.